Page 98 - A Course in Linear Algebra with Applications

82 Chapter Three: Determinants

Proof
In the first place, remember that A~ l exists if and only if
det(A) ^ 0, by 3.3.1. Prom A adj(A) = {det(A))I n we obtain

A(l/det(A))adj(A)) = l/det(A)(A adj(A)) = I n,

by 3.3.6. The result follows in view of 3.3.4.
Example 3.3.2
Let A be the matrix

The adjoint of A is
/ 3 2 1 \

2 4 2 .
\ 1 2 3 /
Expanding det(i4) by row 1, we find that it equals 4. Thus

/ 3 / 4 1/2 1/4 \
A' 1 = 1/2 1 1/2 .
\ l / 4 1/2 3 / 4 /

Despite the neat formula provided by 3.3.7, for matrices with
four or more rows it is usually faster to use elementary row
operations to compute the inverse, as described in 2.3: for
to find the adjoint of an n x n matrix one must compute n
determinants each with n — 1 rows and columns.
Next we give an application of determinants to geometry.

Example 3.3.3

Let Pi(xi, yi, zx), P 2(x 2, y 2, z 2) and P 3 (^3, 2/3, z 3) be three
non-collinear points in three dimensional space. The points